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PreprintarXiv licencearXiv

Weighted Two-Parameter Quantum Divergences Generated by Matrix Power-Mean Gaps

  • Trung Hoa Dinh
  • Minh Toan Ho
  • Ha Trang Nguyen
  • Trung Dung Vuong
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Abstract

Let ℙn be the cone of n× n complex positive definite matrices. For 0<θ<1 and r≥0, let Mr(θ) denote the weighted matrix power mean, with the logarithmic endpoint at r=0, defined by Mr(θ)(A,B) = ((1−θ)Ar+θBr)1/r, & r>0, [2mm] exp((1−θ)log A+θlog B), & r=0. For p>q≥0, we introduce Φp,q(θ)(A,B) = Tr[Mp(θ)(A,B)−Mq(θ)(A,B)]. We prove that Φp,q(θ) is a quantum divergence in the sense of Bhatia-Gaubert-Jain. Moreover, for 0≤ q≤1≤ p≤2, p>q, it is jointly convex and satisfies the data processing inequality for completely positive trace-preserving maps, with the output side interpreted by the regularized boundary convention whenever it is singular. The construction recovers Hellinger-type and Jensen-Shannon-type quantities as special cases or limits. We also compute the induced local quadratic form and provide scalar and matrix obstructions showing that the convexity and data-processing range cannot be extended to all parameters.

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Cite

BibTeXRIS
BibTeX
@misc{dinh2026weighted,
  title = {{Weighted Two-Parameter Quantum Divergences Generated by Matrix Power-Mean Gaps}},
  author = {Trung Hoa Dinh and Minh Toan Ho and Ha Trang Nguyen and Trung Dung Vuong},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06870},
  eprint = {2610.06870},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06870},
}
RIS
TY  - UNPB
TI  - Weighted Two-Parameter Quantum Divergences Generated by Matrix Power-Mean Gaps
AU  - Trung Hoa Dinh
AU  - Minh Toan Ho
AU  - Ha Trang Nguyen
AU  - Trung Dung Vuong
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2610.06870
UR  - https://arxiv.org/abs/2610.06870
AB  - Let ℙ_n be the cone of n× n complex positive definite matrices. For 0<θ<1 and r≥0, let M_r^(θ) denote the weighted matrix power mean, with the logarithmic endpoint at r=0, defined by M_r^(θ)(A,B) = ((1−θ)A^r+θB^r)^1/r, & r>0, [2mm] exp((1−θ)log A+θlog B), & r=0. For p>q≥0, we introduce Φ_p,q^(θ)(A,B) = Tr[M_p^(θ)(A,B)−M_q^(θ)(A,B)]. We prove that Φ_p,q^(θ) is a quantum divergence in the sense of Bhatia-Gaubert-Jain. Moreover, for 0≤ q≤1≤ p≤2, p>q, it is jointly convex and satisfies the data processing inequality for completely positive trace-preserving maps, with the output side interpreted by the regularized boundary convention whenever it is singular. The construction recovers Hellinger-type and Jensen-Shannon-type quantities as special cases or limits. We also compute the induced local quadratic form and provide scalar and matrix obstructions showing that the convexity and data-processing range cannot be extended to all parameters.
ER  -